Chapter 15 Probability Models

The Equally Likely Approach(also called the Classical Approach)

Assigning Probabilities

If an experiment has N outcomes, then each outcome has probability 1/N of occurring

If an event A1 has n1 outcomes, then

P(A1) = n1/N

DiceYou toss two dice. What is the probability of the outcomes summing to 5?

There are 36 possible outcomes in S, all equally likely (given fair dice).

Thus, the probability of any one of them is 1/36.

P(the roll of two dice sums to 5) =

P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111

This is S:

{(1,1), (1,2), (1,3), ……etc.}

We Need Efficient Methods for Counting Outcomes

Counting in “Either-Or” Situations• NCAA Basketball Tournament, 68

teams: how many ways can the “bracket” be filled out?

1. How many games?2. 2 choices for each game3. Number of ways to fill out the bracket:

267 = 1.5 × 1020

• Earth pop. about 6 billion; everyone fills out 100 million different brackets

• Chances of getting all games correct is about 1 in 1,000

Counting Example

Pollsters minimize lead-in effect by rearranging the order of the questions on a survey

If Gallup has a 5-question survey, how many different versions of the survey are required if all possible arrangements of the questions are included?

Solution

There are 5 possible choices for the first question, 4 remaining questions for the second question, 3 choices for the third question, 2 choices for the fourth question, and 1 choice for the fifth question.

The number of possible arrangements is therefore

5 4 3 2 1 = 120

Efficient Methods for Counting Outcomes

Factorial Notation:n!=12 … n

Examples1!=1; 2!=12=2; 3!= 123=6; 4!

=24;5!=120;Special definition: 0!=1

Factorials with calculators and Excel

Calculator: non-graphing: x ! (second function)graphing: bottom p. 9 T I Calculator Commands(math button)

Excel:Insert function: Math and Trig category, FACT function

Factorial Examples20! = 2.43 x 1018

1,000,000 seconds?About 11.5 days1,000,000,000 seconds?About 31 years31 years = 109 seconds1018 = 109 x 109

20! is roughly the age (according to some) of the universe in seconds

Permutations

A B C D EHow many ways can we choose 2

letters from the above 5, without replacement, when the order in which we choose the letters is important?

5 4 = 20

Permutations (cont.)

20)!25(

!5:

45!3

!5

)!25(

!52045

25

PNotation

Permutations with calculator and Excel

Calculatornon-graphing: nPr

Graphingp. 9 of T I Calculator Commands(math button)

ExcelInsert function: Statistical, Permut

Combinations

A B C D EHow many ways can we choose 2

letters from the above 5, without replacement, when the order in which we choose the letters is not important?

5 4 = 20 when order importantDivide by 2: (5 4)/2 = 10 ways

Combinations (cont.)

!)!(

!

102

20

21

45

!2!3

!5

!2)!25(

!525

52

rrn

nC

C

rnnr

ST 305 Powerball Lottery

From the numbers 1 through 20,choose 6 different numbers.

Write them on a piece of paper.

Chances of Winning?

760,38!6)!620(

!20

ies?possibilit ofNumber

important.not order t,replacemen

without 20, from numbers 6 Choose

620206

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Example: Illinois State Lottery

balls) pong pingmillion 16.5 house, ft (1200

months) 10in second 1about (

165,827,25!6!48

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North Carolina Powerball Lottery

Prior to Jan. 1, 2009 After Jan. 1, 2009

:

55!3,478,761

5!50!

:

42!42

1!41!

3,478,761*42

146,107,962

5 from 1- 55

1 from 1- 42 (p'ball #)

:

59!5,006,386

5!54!

:

39!39

1!38!

5,006,386*39

195,249,054

5 from 1- 59

1 from 1- 39 (p'ball #)

Most recent change: powerball number is from 1 to 35http://www.nc-educationlottery.org/faq_powerball.aspx#43

The Forrest Gump Visualization of Your Lottery Chances

How large is 195,249,054?$1 bill and $100 bill both 6” in length

10,560 bills = 1 mileLet’s start with 195,249,053 $1 bills

and one $100 bill …… and take a long walk, putting

down bills end-to-end as we go

Raleigh to Ft. Lauderdale…

… still plenty of bills remaining, so continue from …

… Ft. Lauderdale to San Diego

… still plenty of bills remaining, so continue from…

… still plenty of bills remaining, so continue from …

… San Diego to Seattle

… still plenty of bills remaining, so continue from …

… Seattle to New York

… still plenty of bills remaining, so …

… New York back to Raleigh

Go around again! Lay a second path of bills

Still have ~ 5,000 bills left!!

Chances of Winning NC Powerball Lottery?

Remember: one of the bills you put down is a $100 bill; all others are $1 bills.

Put on a blindfold and begin walking along the trail of bills.

Your chance of winning the lottery is the same as your chance of selecting the single $100 bill if you stop at a random location along the trail and pick up a bill .

More Changes

After Jan. 1, 2009 After Jan. 1, 2012 http://www.nc-educati

onlottery.org/powerball_how-to-play.aspx

:

59!5,006,386

5!54!

:

39!39

1!38!

5,006,386*39

195,249,054

5 from 1- 59

1 from 1- 39 (p'ball #)

Virginia State Lottery

969000,52!1!24

!25760,118,2

760,118,2

760,118,2!5!45

!50: 5Pick

125

550

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